Find a particular solution to the nonhomogeneous differential equation y′′+4y=cos(2x)+sin(2x).

Take the homogeneous part and find the roots to the characteristic equation:



This means the characteristic solution is .

Since the characteristic solution already contains both functions on the RHS of the ODE, you could try finding a solution via the method of undetermined coefficients of the form . Finding the second derivative involves quite a few applications of the product rule, so I'll resort to a different method via variation of parameters.

With and , you're looking for a particular solution of the form . The functions satisfy




where is the Wronskian determinant of the two characteristic solutions.



So you have







So you end up with a solution



but since is already accounted for in the characteristic solution, the particular solution is then



so that the general solution is

Answer

1 number

explanation

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

 

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.